DirectX exporter: does the currently-selected action make a difference?
Short answer: Yes, it does matter.

Long answer: Yes, but not that much. Also, my file was broken for a different reason. The reason the walk action wasn't showing up in the exported file was that it was tied to the Cube mesh as well as a fake user. If the walk action had only one user (a fake one), then the "Attach Unused Actions to First Armature" option would have properly created this animation in the .x file and attached it to the Armature. 

In my file, the walk action was not unused because it was tied to the mesh. The exporter created an animation for it and attached it to the mesh, not the Armature. This action took up fewer lines in the .x file because there were no keyframes on the mesh itself, so the exporter didn't export any.

The moral of the story:


Make sure your actions all have fake users
Make sure your actions don't have any extraneous users